Normality of Cayley (di)graphs has also been investigated intensively. Godsil [1]} proved that if \(\hbox{\rm Cay}(G,S)\)  is normal then \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\)  is a semidirect product of \(R(G)\)  by the subgroup \(\hbox{\rm Aut}(G,S):=\lbrace \alpha \in \hbox{\rm Aut}(G)\ |\ S^\alpha =S\rbrace \)  (also see Proposition REF ), and hence \(\hbox{\rm Aut}(\hbox{\rm Cay}(G,S))\)  is completely determined by \(\hbox{\rm Aut}(G)\) . Wang et al. [2]} obtained that every finite group \(G\)  has a normal Cayley graph unless \(G\cong C_4 \times C_2\)  or \(G \cong {Q}_8 \times C_2^m\) , where \(C_n\)  stands a cyclic group of order \(n\)  and \({Q}_8\)  is the Quaternion group. The normality of Cayley graphs of cyclic groups of order a prime and of groups of order twice a prime was solved by Alspach [3]} and Du et al. [4]}, respectively. Dobson [5]} determined all non-normal Cayley graphs of order a product of two distinct primes, and Dobson and Witte [6]} determined all non-normal Cayley graphs of order a prime-square. For normality of Cayley graphs of finite simple groups, we refer the reader to [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]},
and for some results on normality of Cayley digraphs, one may see [19]}, [20]}, [21]}, [22]}.
Based on these results, Xu [23]} conjectured that almost all connected Cayley graphs are normal.
